of transformation-induced plasticity
in multiphase carbon steels
als Research (NIMR) and the Stichting voor Fundamenteel Onderzoek der
Ma-terie (FOM), financially supported by the Nederlandse organisatie voor Weten-schappelijk Onderzoek (NWO). The research is carried out under project number
02EMM20 of the FOM/NIMR program “Evolution of the Microstructure of Ma-terials” (P-33).
of transformation-induced plasticity
in multiphase carbon steels
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 29 januari 2008 om 10 uur
door
Denny Dharmawan TJAHJANTO
ingenieur toegepaste wiskunde geboren te Cirebon, Indonesi¨e
Prof. dr. ir. S. van der Zwaag Toegevoegd promotor: Dr. S.R. Turteltaub
Samenstelling promotiecommissie: Rector Magnificus, Voorzitter
Prof. dr. ir. S. van der Zwaag, Technische Universiteit Delft, promotor
Dr. S.R. Turteltaub, Technische Universiteit Delft, toegevoegd promotor Prof. dr.-ing. D. Raabe, Max-Planck-Insitut f¨ur Eisenforschung
Prof. dr. ir. M.G.D. Geers, Technische Universiteit Eindhoven Prof. dr. ir. T. Pardoen, Universit´e Catholique de Louvain Prof. dr. ir. L.J. Sluys, Technische Universiteit Delft Dr. ir. A.S.J. Suiker, Technische Universiteit Delft
Dr. ir. A.S.J. Suiker heeft als begelieder in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
Trefwoorden:
Martensitic transformation, Crystal plasticity, Transformation-induced plasticity, Thermo-mechanical framework, Consistent stress-update algorithm, Finite ele-ment method, Homogenization scheme, Microstructural properties
Copyright c 2007 by D.D. Tjahjanto
Printed in the Netherlands by PrintPartner Ipskamp ISBN-13: 978-90-9022499-2
This thesis summarizes the four-year research project I have done on the design of optimized multiphase transformation-induced plasticity (TRIP)-assisted steels. The work is part of a joint research program between the Netherlands Institute for Metals Research (NIMR) and the Stichting Fundamenteel Onderzoek der
Ma-terie (FOM), which is financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The research is performed under project
number 02EMM20 of the FOM/NIMR program “Evolution of the Microstructure of Materials” (P-33).
First of all, I would like to gratefully acknowledge Prof. Sybrand van der Zwaag as the promotor for the effective support and guidance during this re-search, and Dr. Sergio Turteltaub and Dr. Akke Suiker, who have provided an excellent day-to-day supervision and many inspirations. In addition, I would like to acknowledge Dr. Pedro Rivera for all discussions and feedbacks on the thermo-dynamical and metallurgical aspects of the models, and Prof. Ren´e de Borst for the opportunity to use the research facilities in the Engineering Mechanics (EM) group. Furthermore, I would like to thank Prof. Dierk Raabe, Dr. Franz Roters and Dr. Philip Eisenlohr for offering me a wonderful place during a three-month visit to the Max-Plank-Institut f¨ur Eisenforschung (MPI-E) in D ¨usseldorf, and for their assistance during this visit.
Next, I would like to express my gratitude to Prof. Marc Geers (Eindhoven University of Technology), Prof. Thomas Pardoen (Universit´e Catholique de Lou-vain) and Prof. Bert Sluys (Delft University of Technology) as the members of the doctoral committee, as well as to Prof. Gijs Ooms (Delft University of Technol-ogy) as the reserve member. Furthermore, I would like to acknowledge the discus-sions with Prof. John Bassani (University of Pennsylvania) on the basic concept and the implementation of the non-glide stress effect in BCC crystals. In addition, I address my gratitute to the fellow researchers in the NIMR Cluster 5 and to the Corus Research Development and Technology (RD&T) team for the discussions
and feedback during this research.
I owe many thanks to Carla Roovers, Harold Thung and Laura Chant for the wonderful assistance to solve administrative and technical issues. In addition, I am indebted to all colleagues and former colleagues at the EM group (Prof. Miguel Guti´errez, Dr. Steven Hulshoff, Dr. Harald van Brummelen, Dr. Christian Mich-ler, Dr. Edwin Munts, Dr. DooBo Chung, Dr. Olaf Herbst, Thomas Hille, Clemens Verhoosel, Andr´e Vaders, Marcela Cid, Juliana Lopez, Wijnand Hoitinga, Gertjan van Zwieten, Jingyi Shi, Kris van der Zee and Ido Akkerman) and at the Funda-mentals of Advanced Materials (FAM) group (Dr. David San Martin and Dr. Doty Risanti) for creating a pleasant atmosphere and interesting discussions.
Personally, I would like to deeply thank Angelica Tanisia, Fr. Ben Engel-bertink, Rev. Waltraut Stroh and Kasia “my virtual sister” Wac for all support, courage and motivation that were given during the last couple of years. I am also grateful to the Indonesian community in Delft (particularly, Julius Sumihar, Ferry Permana, Sinar Juliana, Dwi Riyanti, Xander Campman, Nelson Silitonga, Iwan Kurniawan, Henri Ismail, Sandy Wirawan and Yuli Tanyadji) and friends in the International Student Chaplaincy Delft (especially, Ruben Abellon, Fr. Avin Kunnekkadan, Francesca Mietta, Carmen Lai, Ludvik Lidicky, Anna Dall’Acqua, Maria Parra, Henk van der Vaart and Mieke and Reini Knoppers) for sharing a lot of fun during my stay in Delft. Last but not least, I would like to thank my family and friends in Indonesia, for their long-distance support and prayers.
All in all, I wish that this thesis gives valuable knowledge and insight to all people interested in studying the TRIP effect in steels. Enjoy reading!
Contents ix
1 Introduction 1
1.1 Background: Multiphase TRIP-assisted steels . . . 2
1.1.1 Two-stage heat-treatment process for TRIP steels . . . 3
1.1.2 Martensitic transformation in low-alloyed carbon steels . . 4
1.1.3 Microstructural parameters influencing the stability of austen-ite against transformation . . . 5
1.1.4 Modeling of TRIP effect in steels: State of the art . . . 6
1.2 Objectives and scope . . . 7
1.3 Thesis outline . . . 8
1.4 General scheme of notation . . . 9
2 Elasto-plastic deformation of single-crystalline ferrite 11 2.1 Single crystal elasto-plastic model for ferrite . . . 13
2.1.1 Kinematics and configurations . . . 14
2.1.2 Thermodynamic formulations . . . 16
2.1.3 Constitutive relations and Helmholtz energy density . . . 20
2.1.4 Driving force, non-glide stress and kinetic law . . . 24
2.1.5 Hardening and evolution of microstrain . . . 25
2.2 Simulations of elasto-plastic deformation of single-crystalline ferrite 29 2.2.1 Material parameters and validation . . . 29
2.2.2 Sample geometry and boundary conditions . . . 31
2.2.3 Stress-strain response of single-crystalline ferrite . . . 33
3 Elasto-plastic-transformation behavior of single-crystalline austenite 47 3.1 Single crystal elasto-plastic-transformation model for austenite . . 49
3.1.2 Thermodynamic formulations . . . 53
3.1.3 Constitutive relations and Helmholtz energy density . . . 57
3.1.4 Driving forces, nucleation criteria and kinetic laws . . . . 64
3.1.5 Hardening and evolution of microstrain . . . 67
3.2 Simulations of elasto-plastic-transformation behavior of single-crystalline austenite . . . 71
3.2.1 Material parameters and validation . . . 71
3.2.2 Sample geometry and boundary conditions . . . 75
3.2.3 Stress-strain response of single-crystalline austenite . . . . 77
4 Numerical solution algorithm for transformation-plasticity model 89 4.1 Stress-update algorithm for coupled transformation-plasticity model 90 4.1.1 Discretization of model equations . . . 91
4.1.2 Newton-Raphson iteration procedure (return-mapping) . . 96
4.1.3 Consistency checks for slip and transformation systems . . 100
4.1.4 Sub-stepping procedure . . . 102
4.2 Tangent operator . . . 104
4.2.1 Finite difference approximation for tangent operator . . . 105
4.2.2 Tangent operator in the Eulerian setting . . . 105
4.3 Validation of the numerical solution algorithm . . . 107
4.3.1 Sample geometry and finite element meshes . . . 107
4.3.2 Simulation results (mesh refinement analysis) . . . 108
5 Micromechanical simulation of TRIP-assisted steel 113 5.1 Simulation of multiphase TRIP steel at single grain level . . . 114
5.1.1 Microstructural sample geometry and boundary conditions 114 5.1.2 Strain-strain response of TRIP steel microstructure . . . . 117
5.2 Parametric study of polycrystalline TRIP steel behavior as a func-tion of microstructural properties . . . 127
5.2.1 Sample geometry and boundary conditions . . . 128
5.2.2 Microstructural configuration and model parameters . . . 128
5.2.3 Simulation results . . . 133
6 Macroscale simulation of multiphase TRIP-assisted steels 141 6.1 Homogenization scheme for multiphase microstructure . . . 143
6.1.2 Preliminary analysis and comparison to direct FEM
sim-ulation . . . 145
6.2 Deep-drawing simulation of multiphase TRIP-aided steel . . . 149
6.2.1 Sample geometry and boundary conditions . . . 149
6.2.2 Sample crystallographic orientation distribution function . 151 6.2.3 Simulation results and analysis . . . 155
7 Simulation of thermal behavior of multiphase TRIP-assisted steel 161 7.1 Single-crystalline thermo-mechanical models for multiphase TRIP-assisted steel . . . 162
7.1.1 Thermo-elasto-plastic-transformation model for austenite . 163 7.1.2 Thermo-elasto-plasticity model for ferrite . . . 166
7.2 Simulation of TRIP steel behavior under cooling . . . 167
7.2.1 Boundary conditions and model parameters . . . 167
7.2.2 Analysis of TRIP steel behavior under cooling . . . 170
7.2.3 Comparison with experimental results . . . 177 A Kinematics of martensitic transformation at lower length-scales 183
B Effective elastic stiffness for martensitic transformation systems 187
C Plastic slip systems for FCC austenite and BCC ferrite 189
References 193
Summary 209
Samenvatting 213
Intisari 217
1
Introduction
The improvement of strength in carbon steels is often obtained at the expense of ductility, and vice-versa. A long-standing ambition has been to develop a class of steels where both ductility and strength can be simultaneously improved. For this purpose, transformation-induced plasticity (TRIP)-assisted steels are particulary appealing. TRIP-assisted steels are a class of multiphase steels that exhibit a good combination of strength and ductility characteristics. This unique characteristic is attributed to the presence of a metastable austenitic phase in the microstruc-ture at room temperamicrostruc-ture. Upon applied thermal and/or mechanical loadings, the metastable retained austenite may transform into a harder martensitic phase, which may increase the effective strength of the material. In addition, transforma-tion from austenite to martensite is accompanied by shape and volume changes, which are accommodated by local plastic deformations in the surrounding phases, creating the so-called “TRIP-effect” [61]. The additional plastic deformation due to the transformation increases the effective work-hardening of the material. In comparison to similar steels that contain no retained austenite in their microstruc-ture, e.g., dual-phase (DP) steels, TRIP-assisted steels have a similar (ultimate) strength, but exhibit a significantly higher ductility.
1.1
Background: Multiphase TRIP-assisted steels
At room temperature, a typical microstructure of TRIP steel consists of several phases, i.e., the intercritical ferrite as the most dominant phase, bainite, retained austenite and occasionally a small fraction of thermal martensite [8, 34, 42, 64, 102, 112, 133]. Intercritical ferrite (sometimes also referred to as pro-eutectoid ferrite) occupies up to 75 % volume of the microstructure. Ferrite has a
body-centered cubic (BCC) lattice and, compared to other constituent phases, is the softest phase. Nano-indentation tests performed by Furn´emont et al. [42] showed that the hardness of ferrite in a typical multiphase steel is about 5 GPa. As
re-ported in the literature [42, 64], the size of ferritic grains in a typical TRIP steel microstructure ranges from5 to 10 µm. Unlike ferrite, bainite does not have a
single-phase structure. The microstructure of bainite consists of an assembly of layers of iron carbide (cementite) and bainitic ferrite. Bainite is formed during an isothermal bainitic holding at a temperature between600 and 700 K. In
gen-eral, bainite is harder than intercritical ferrite due to its smaller grain size and the presence of carbide precipitations. The typical size of bainitic grains ranges from
1 up to 6 µm. In addition, initial bainite can possess a higher dislocation
den-sity [64]. In the case of TRIP steels, the chemical composition is chosen such that the formation of carbides is restricted (or postponed), which results in a bainite in TRIP steels that is essentially carbon-free, but still has the characteristics of a fine plate-like structure [42, 61, 64]. The next constituent phase in TRIP steel microstructure is retained austenite. In contrast to other constituent phases that are stable, retained austenite is a metastable phase. In general, austenite is a high temperature phase, which has a face-centered cubic (FCC) structure. Stabilization of austenite at room temperature is due to local carbon enrichment and the con-straining effect from neighboring grains. Upon the application of thermal and/or mechanical loads, metastable austenite may transform into martensite and gener-ate the TRIP effect. In some cases, the initial TRIP steel microstructure may also contain a small fraction of thermal martensite. Thermal martensite is obtained when austenite is rapidly cooled (or quenched) such that diffusion of carbon is pre-vented during transformation. Martensite has a body-centered tetragonal (BCT) structure that contains supersaturated interstitial carbon atoms, which can create strain fields that restrict the movement of dislocations in the lattice [23]. Marten-site can also have a high dislocation density resulting from a displacive (or diffu-sionless) transformation mechanism. In comparison to other constituent phases,
Te m per ature θ Δt2 Time t Δt1 θ1 θ2 Intercritical annealing Bainitic holding Quenching A F A F B A B F =Austenite =Bainite =Ferrite
Figure 1.1: Schematic representation of temperature profile of the two-stage heat treatment typically used in low-alloyed TRIP steels processing and the corre-sponding microstructural phases obtained at the end of each stage.
martensite shows the highest hardness level. Nano-indentation tests by Furn´emont
et al. [42] indicated that the hardness of martensite can exceed17 GPa. In TRIP
steel microstructures, martensite appears in platelets or needle-shaped laths.
1.1.1 Two-stage heat-treatment process for TRIP steels
In many cases, the microstructure of multiphase TRIP-assisted steels are produced through a two-stage heat treatment process [59–61, 64, 92, 102, 133]. Similar to the processing route for dual-phase (DP) steels, the first stage of the heat treat-ment process is the intercritical annealing, in which the material is brought to a temperature θ1 between the intercritical temperatures A1 and A3. This process
transforms some parts of the initial microstructure into the austenitic phase. As a result, the microstructure after the intercritical annealing process consists of two phases, i.e., (pro-eutectoid) ferrite and austenite, as schematized in Figure 1.1. While for dual-phase steels the microstructure resulting from the intercritical an-nealing is directly quenched to a room temperature, the intermediate TRIP steel microstructure is brought to a bainitic temperatureθ2for isothermal holding over
a period∆t2. During this isothermal holding, a fraction of the austenite formed
during intercritical annealing transforms into bainite, whereas the remaining part of the austenite is further stabilized by the enrichment of carbon expelled from the bainite formed. At this stage, the size of the retained austenite grains in the
Austenite
Martensite
Thermal (Gibbs) fre
e en er g y G ΔGA→M(θ) < Gbarrier Temperature θ θT ΔGA→M(Ms) = Gbarrier GA GM θ Ms Gmech
Figure 1.2: Schematic representation of thermal (or chemical) free energy of the austenitic and martensitic phases as functions of temperature.
resulting microstructure depends on the bainitic holding temperature T2 and the
holding time∆t2. After the isothermal bainitic holding, the steel is quenched to
room temperature. In general, the phase composition created during the bainitic holding is preserved during the final quenching. Occasionally, a small fraction of retained austenite further transforms into thermal martensite during the final quenching, particularly in the austenitic regions in which the carbon enrichment was not sufficient.
1.1.2 Martensitic transformation in low-alloyed carbon steels
Martensitic transformations occur as a consequence of energy minimization among the phases in the microstructure. At high temperatures, the austenitic phase pos-sesses a lower free-energy level than the martensitic phase and, therefore, is a stable phase. Conversely, at low temperature, the martensitic structure becomes more favorable since it has a lower free-energy level. Figure 1.2 schematically illustrates the free-energy of the austenitic and martensitic phases as functions of temperature. Transformation from austenite to martensite in carbon steels can be triggered either by thermal loading (cooling) or through the application of external mechanical loading. Upon cooling and in the absence of stress, the transformation from austenite to martensite starts to occur at the transformation temperature,Ms,
en-ergy”) between the austenite and martensite,∆GA→M, is sufficient to overcome
the transformation energy barrier (Gbarrier).
At temperatures higher thanMs, transformation may occur with the assistance
of mechanical stress, such that the mechanical strain energy (Gmech) added to the
thermal energy difference is sufficient to overcome the transformation barrier, as shown in Figure 1.2. However, in carbon steels, stress-assisted martensitic trans-formation are irreversible, i.e., reverse transtrans-formations (from martensite to austen-ite) cannot occur upon reversal of loading. Transformation from martensite to austenite can only be realized by re-heating. This is in contrast to shape-memory alloys, where stress-assisted transformations are crystallographically reversible.
1.1.3 Microstructural parameters influencing the stability of austen-ite against transformation
In the final microstructure, the stability of retained austenite grains against trans-formation plays an important role in characterizing the overall performance of TRIP-assisted steels. Experimental investigations have shown that the stabil-ity of the austenitic grains is influenced by various microstructural parameters, such as (i) the carbon concentration in the retained austenite [8, 34, 64, 102, 112], (ii) the size and shape of the austenitic grains [11, 68, 147], (iii) the morphol-ogy of microstructural phases [63, 64, 151], (iv) the crystallographic orientation of grains (microstructural texture) [72, 91] and (v) the stiffness of the surrounding phases [102, 133].
In TRIP steel microstructures, the volume fraction of the phases, the size and shape of the austenitic grains, as well as their local carbon concentration obtained in the two-stage heat treatment process depend upon the intercritical annealing and bainitic transformation process conditions, e.g., temperature and holding time [59, 60, 64, 84, 102, 112, 147]. For example, a longer bainitic hold-ing time results in a final microstructure with smaller grains of retained austenite, but with a higher carbon content. Furthermore, the carbon concentration in the retained austenite grains is controlled through the presence of alloying elements, such as silicon, aluminum and phosphor [12]. These elements effectively pre-vent carbide precipitation during the bainitic holding stage and, thus, enhance the carbon enrichment in the austenite. For typical TRIP steel microstructures, the carbon concentration in the retained austenite reportedly varies from0.6 wt.% up
deter-mination of the carbon concentration in the austenitic grains is rather complex and there is no generally accepted method for determining the austenite carbon concentration. It is generally accepted that the real range of austenite carbon con-centrations in multiphase TRIP steels is probably smaller than first indicated.
Besides carbon enrichment, the stability of retained austenite is influenced by the mechanical properties of the surrounding phases. Under external thermo-mechanical loading the stresses experienced by the austenitic grains depend on the elasto-plastic properties of the surrounding ferritic grains (e.g., yield stress and strain hardening behavior), which can be controlled by the addition of ele-ments, such as manganese and molybdenum, as well as by changing the ferritic grains size [43, 66]. In addition, the crystallographic orientation of the austenitic grains with respect to the loading direction plays an important role in the austen-ite stability against transformation [72, 91]. This information is relevant for the macroscopic behavior of TRIP steels, particulary if the steel is produced through a rolling process, which may induce a microstructural texture, where a large num-ber of grains are oriented in a specific crystallographic direction [144, 149].
1.1.4 Modeling of TRIP effect in steels: State of the art
The modeling of the TRIP effect involves two key aspects [39, 64], namely (i) the elasto-plastic deformation in the transforming austenitic region as well as in the neighboring phases to accommodate shape and volume changes associated with the martensitic transformation, which is often referred to as the
Greenwood-Johnson effect [45] and (ii) the strong dependency of the martensitic formation
upon the crystallographic orientation with respect to the loading axis, also known as the Magee effect [83]. From a historical point of view, the modeling of marten-sitic phase transformations can be traced back to the pioneering work of Wechsler
et al. [148] in 1953, where a crystallographically-based model was proposed to
de-scribe the kinematics for a martensitic transformation. This concept was refined by Ball and James [7] by formulating the model within an energy minimization framework. During the last decades, various constitutive models for martensitic transformations have been developed for describing the TRIP effect, such as the one-dimensional phase transformation model of Olson and Cohen [93], which was extended into a three-dimensional model by Stringfellow et al. [120] and Bhattacharyya and Weng [16]. Furthermore, models based on a more complex micromechanical framework were also proposed. These can be found in the work
of, e.g., Leblond et al. [75, 76] and Levitas et al. [79, 80]. Following the classical crystallographic model of Wechsler et al. [148], Marketz and Fischer [85, 86] pro-posed a model for stress-assisted martensitic transformation for single-crystalline and polycrystalline austenite, see also Tomita and Iwamoto [138, 139]. Further, Diani et al. [32, 33] proposed a model that takes into account the effect of the crystallographic orientations of grains on the elasto-plastic response using a small strain formulation. Similar models can be also found in Cherkaoui et al. [24, 25] and Taleb and Sidoroff [127].
Most of the models mentioned above were developed within a small-strain framework, which can lead to inaccurate predictions since martensitic transfor-mations can locally induce large elasto-plastic defortransfor-mations, even if the effec-tive macroscopic deformation is relaeffec-tively small. In addition, an isotropic elasto-plastic response is often assumed, which is a strong simplification, particulary for analyses at smaller length scales (e.g., at the single-crystal level), where the effect of anisotropy due to crystallographic orientations cannot be neglected [41, 72, 91]. Within the context of a large deformation framework, Turteltaub and Suiker [124, 141, 143] have developed a crystallography-based model for martensitic phase transformations in carbon steels. The model is derived following a multiscale ap-proach, where material parameters, e.g., transformation deformation kinematics and effective elastic stiffness, at higher length-scales are calculated from lower scale quantities by means of averaging schemes. In addition, the model is con-structed within a thermo-mechanically consistent framework, where the thermal quantities are derived analogous to the mechanical counterparts. The model of Turteltaub and Suiker [124, 141, 143] lays the foundation for the work to be pre-sented in this thesis.
1.2
Objectives and scope
Despite of its superior characteristics, there is room for further improvement in the overall performance of a TRIP-assisted steel. However, this can only be achieved by developing a thorough understanding of the TRIP mechanism. The present work is aimed at developing crystallographically-based computational models for simulating the behavior of TRIP-assisted steels. The underlying goal of the sim-ulations is to study systematically the mechanism of TRIP, particularly the mech-anism of the stress-assisted martensitic transformation in the austenitic grains, as well as the elasto-plastic interactions between the transforming grains and the
neighboring phases. The models developed to study these effects are numerically implemented within a finite-element framework.
Within a parametric analysis setting, various sets of simulations are performed in order to identify the role of every microstructural property on the austenite sta-bility and, thus, the overall response of the TRIP-assisted steels under mechanical and thermal loadings. The analyses cover various length-scales, i.e., from simu-lations at the level of single crystal up to simusimu-lations of forming processes at the macroscopic scales. On the whole, the present work will provide a good insight for further improvement of the performance of TRIP-assisted steels, as well as for the optimization of the TRIP steel processing parameters.
1.3
Thesis outline
The outline of this thesis is as follows: The elasto-plastic responses of single-crystalline ferrite are simulated and studied in Chapter 2. For this purpose, a single crystal elasto-plasticity model is adopted. In order to mimic the asym-metric behavior of slip in twinning and anti-twinning senses typically found in BCC metals, the model incorporates the effect of the non-glide stress into the kinetic formulation. The model is derived within a large deformation frame-work. In order to demonstrate the basic features of the model, several simula-tions are performed for various types of elementary deformation modes. Sub-sequently, a crystallography-based model for simulating the behavior of single-crystalline austenite is presented in Chapter 3. This model is derived through coupling the multiscale martensitic phase transformation model of Turteltaub and Suiker [141, 143] to an FCC single-crystal elasto-plasticity model. The coupling between the transformation and the plasticity terms is derived systematically using a thermodynamically-consistent formulation. The model is used to study the re-sponse of single-crystalline austenite, in particular the interaction between phase transformation and plastic deformation mechanisms in the austenite under various loading conditions.
Key aspects of the numerical implementation of the models are presented in Chapter 4. The discussion is thereby mainly focussed on the numerical imple-mentation of the elasto-plastic-transformation model for the austenitic phase. The numerical algorithm for the ferrite elasto-plasticity model can be performed anal-ogously through eliminating the terms related to transformation. In addition, a number of simulations are presented to show the numerical stability and
conver-gence of the implemented algorithm.
In Chapter 5, the single-crystalline models presented in Chapters 2 and 3 are combined to simulate the response of multiphase TRIP-assisted steel microstruc-tural samples. The simulations focus on the interaction between the transforming austenitic grain and the surrounding ferritic matrix for different combinations of crystallographic orientations. In addition, the role of microstructural properties, such as local carbon concentration and austenitic grain size on the overall re-sponses of the TRIP steels are studied in a parametric analysis. In Chapter 6, simulations of TRIP steel behavior at the macroscopic scale, e.g., during deep-drawing process, are shown. For this purpose, the present single-crystalline mod-els for ferrite and austenite are employed in combination with a simple averaging scheme, namely the iso-work-rate weighted-Taylor scheme. A direct reconstruc-tion of orientareconstruc-tion distribureconstruc-tion funcreconstruc-tions (ODF) by means of a probabilistic ap-proach is performed in order to replicate the crystalline texture of the samples during simulations. Finally, the behavior of multiphase TRIP-assisted steels dur-ing thermal loaddur-ing is simulated and analyzed in Chapter 7. The analyses cover the thermal behavior of TRIP-assisted steels as a function of microstructural prop-erties, similar to the analyses of the mechanical loading presented in Chapter 5. Moreover, the transformation behavior under thermal loading as predicted by the present models is compared to experimental observations.
1.4
General scheme of notation
As a general scheme of notation, scalars are written as lightface italic letters, vectors as boldface lowercase letters (e.g., a, b), second-order tensors as boldface capital letters (e.g., A, B) and fourth-order tensors as blackboard bold capital letters (e.g., A, B). For vectors and tensors, Cartesian components are denoted as
ai, Aij andAijkl. The action of a second-order tensor upon a vector is denoted
as Ab (in components Aijbj, with implicit summation on repeated indices) and
the action of a fourth-order tensor upon a second order tensor is designated as
AB (i.e.,AijklBkl). The composition of two second-order tensors is denoted as
AB (i.e., AijBjl). The tensor product (dyadic product) between two vectors is
denoted as a⊗ b (i.e., aibj). All inner products are indicated by a single dot
between the tensorial quantities of the same order, e.g., a· b for vectors and A · B for second-order tensors (in components, respectively, aibi and AijBij). The
−1. A superimposed dot represents a material time derivative. Subscripts A, M,
andF indicate that the quantities correspond to material properties of austenite,
2
Elasto-plastic deformation of
single-crystalline ferrite
In multiphase steels assisted by the TRIP effect, ferrite is the most dominant phase in terms of its volume. Therefore, although it is not considered as the most important ingredient of TRIP steel microstructure, the elasto-plastic behav-ior of the ferrite-based matrix determines to a large extent the overall behavbehav-ior of the multiphase steel. Despite of this, little attention has been given to the mod-eling of ferrite in many models for TRIP-assisted steels available in the litera-ture [57, 58, 109, 110, 120], where relatively simple elasto-plasticity models were used for the non-transforming phase.
Continuum models used to simulate the elasto-plastic behavior at the level of a single crystal are often based on crystal plasticity theory. The plastic deformation is kinematically described as the result of slip on specific crystallographic planes and in specific directions (i.e., slip systems). The foundations of crystal plastic-ity theory were laid down in the works of Taylor and Elam [129, 130] and Tay-lor [128]. The concept was further developed by Rice [108], Hill and Rice [53], Asaro and Rice [6], Asaro and Needleman [5], Peirce et al. [100, 101] Bassani and Wu [10], Cuiti˜no and Ortiz [30], Gurtin [46] and Gurtin and Anand [48, 49].
Slip-rate and hardening constitutive relations in some of these models, particu-larly the early ones, have a strong phenomenological nature, whereas more recent models attempt to connect the slip mechanism to smaller length scale phenomena through dislocation-based constitutive relations.
In general, crystal plasticity models based on the classical Schmid stress (re-solved shear stress) provide satisfactory predictions for crystalline materials with close-packed structures, such as face-centered cubic lattices (FCC). However, phases with body-centered cubic lattices (BCC), such as ferrite, require additional attention for the following reasons: (i) Due to the absence of close-packed planes, there is no clear definition of slip systems in BCC structures. The lack of ex-perimental observations with sufficient resolution has precluded reaching consen-sus among researchers regarding the crystallographic slip planes that are active during plastic flow. This controversy has been aggravated since experimental ob-servations indicate that the trace of slips in BCC metals also depends on tem-perature [99, 113]; (ii) as opposed to FCC lattices, slip in BCC lattices behaves asymmetrically in the twinning and antitwinning directions [36, 54, 55], which, at macroscopic scales, results in an asymmetric response in tension and compres-sion.
Several crystal plasticity-based models have been proposed for BCC metals within single-crystalline and polycrystalline contexts, for example, Nemat-Nasser
et al. [90], Stainier et al. [118], Peeters et al. [97, 98] and Ma et al. [82]. Those
models shared some similarities, e.g., the above models include the families of
{211} and {321} planes as potential slip planes in order to solve the ambiguity
of slip traces in BCC metals. On the other hand, the issues of asymmetric behav-ior of slip in the twinning and antitwinning directions is, unfortunately, not well addressed. Since the model for BCC single crystals proposed here is part of a bigger framework of TRIP steel modeling, the accuracy on the prediction of the ferritic stress-strain behavior, particularly the asymmetry in tension-compression, is important. In this chapter, a thermodynamically-consistent elasto-plastic model for BCC ferrite is developed. The present model is based on the non-glide stress formulation proposed by Bassani et al. [9], which allows to predict the twinning-antitwinning asymmetric behavior in BCC crystals. The formulation of the elasto-plasticity model for ferrite single crystal is discussed in Section 2.1. Simulations of single-crystalline ferrite with elementary loading modes are presented in Sec-tion 2.2 in order to illustrate the key features of the present model.
2.1
Single crystal elasto-plastic model for ferrite
Plastic slip in BCC crystals operates through propagation of screw dislocations along theh111i directions. Due to the lack of close-packed planes in BCC
struc-tures, there is no clear consensus among researchers on which planes crystal-lographic slip occurs. Experimental observations often indicate slip traces along
{110}, {211} and {321} planes, and sometimes along non-crystallographic planes.
However, higher resolution micrographs show that slip on{321} and higher index planes appear in small wavy patterns, which can be interpreted as slip composed of alternating glide contributions along lower index planes [40, 54, 106]. Although slip traces along{211} planes in BCC ferrite have been reported, particularly at and above room temperature, with the addition of silicon, which is the case for the ferritic matrix in TRIP-assisted steels, it is observed that slip occurs predom-inantly along the {110} planes [113]. Furthermore, atomistic simulations per-formed by Vitek and co-workers [145, 146] indicate that slip along {211} planes can be constructed of equal segments of slip along alternating {110} planes.
Ac-cordingly, it will be assumed in the present formulation that the systems corre-sponding to the{110}h111i family are sufficient to describe slip in ferrite.
The classical approach in crystal plasticity theory is to assume that gliding along an individual slip system is solely determined by the Schmid law, in which the resolved shear stress is equated with the corresponding critical value represent-ing resistance against slip. Although this assumption works reasonably well for FCC metals, it cannot be directly applied to BCC crystals. Atomistic simulations of BCC crystals performed by Duesbery and Vitek [36] have shown that for met-als with a BCC structure, the cores of 12h111i screw dislocations spread into three {110} planes intersecting along the h111i directions. The non-planar spreading
of a dislocation core causes the slip along an individual plane of the{110} class
to become dependent on resolved stresses acting on or normal to another {110}
plane of the [111] zone, called the “non-glide plane”. The resolved stress
act-ing on or normal to the non-glide plane is referred to as non-glide stress. In the present model, the effect of non-glide stress is incorporated following the ap-proach developed by Bassani et al. [9, 145]. Although the non-glide stress model was originally derived based on atomistic simulations of BCC molybdenum and tantalum, it is assumed that the plastic slip in BCC ferrite can be described by a similar mechanism. This assumption is reasonable since the ferrite lattice shares the same generic features with the lattice of molybdenum and tantalum, e.g. the
X mF nF x Fp Fe Reference configuration Current configuration Intermediate (or relaxed) configuration
F(x)
Infinitesimal neighborhood of x
y(x)
Figure 2.1: Schematic representation of the decomposition of deformation gradi-ent F . Vectors mF and nF are, respectively, the slip direction and the slip plane
normal of BCC ferrite in the intermediate (relaxed) configuration [47].
asymmetry of slip in twinning and antitwinning directions [55]. The validation of this assumption would require atomistic simulations of deformation of BCC ferrite; however these fall outside of the scope of the present work.
2.1.1 Kinematics and configurations
Based on a large deformation theory, the total deformation gradient F is decom-posed as [6, 53, 77]
F = FeFp, (2.1)
where Fe is the elastic part of the total deformation gradient F , describing the
deformation due to elastic distortion of the lattice, and Fpis the plastic part of the
total deformation gradient representing deformation related to cumulative crys-tallographic slips. It is assumed that the plastic part of the deformation gradient,
Fp, does not change the lattice structure and that the elastic properties of the ma-terial remain unaltered during a deformation process. As shown in Figure 2.1, the decomposition of the total deformation gradient can be illustrated through the introduction of a reference configuration, an intermediate (relaxed) configuration and a current configuration. The plastic deformation gradient Fp maps a material
In turn, the elastic deformation gradient Feprojects the point from the
intermedi-ate configuration to the current configuration. It is worth noting that, in general, the order of the decomposition does not correspond to the actual deformation se-quence and that the elastic and plastic parts do not correspond to the gradients of globally-defined functions [47]. Furthermore, although the present model is derived to study coupled thermo-mechanical problems, the decomposition of the total deformation gradient in (2.1) does not include the effect of thermal expan-sion/contraction. In this chapter, it is assumed that thermal expansion/contraction is relatively small, and thus, may be neglected. Nonetheless, the incorporation of the thermal expansion/contraction in the total deformation gradient will be dis-cussed in Chapter 7.
The velocity gradient in the current configuration, denoted as ˜L, can be written
as the sum of the elastic part ˜Leand the plastic part ˜Lp, i.e.,
˜
L= ˙F F−1 = ˙FeFe−1+ FeF˙pFp−1F −1
e := ˜Le+ ˜Lp. (2.2)
Note that the velocity gradients ˜Leand ˜Lpare measured in the current
configura-tion. In the intermediate (relaxed) configuration, the plastic velocity gradient Lp
is determined by the cumulative slip rates on all possible slip systems as
Lp:= ˙FpFp−1 = NF
X
i=1
˙γF(i)m(i)F ⊗ n(i)F , (2.3)
where ˙γF(i) is the rate of slip on a system i and the vectors m(i)F and n(i)F are, respectively, unit vectors describing the slip direction and the normal to the slip plane of the corresponding system in the intermediate configuration. In view of (2.3), the rate of change in volume due to plastic slip is given by
d(det Fp)
dt = det Fptr Lp= det Fp
NF
X
i=1
˙γF(i)m(i)F · n(i)F = 0 , (2.4)
where the last relation follows from the fact that the vectors m(i)F and n(i)F are orthogonal to each other for all slip systems, i.e., m(i)F · n(i)F = 0. Consequently,
if the initial plastic deformation gradient is such that det Fp(0) = 1, it follows
thatdet Fp(t) = 1 for all t∈ [0, T ], i.e., the plastic deformation is isochoric.
as ˜ L= ˜Le+ NF X i=1
˙γ(i)F m˜(i)F ⊗ ˜n(i)F , (2.5)
where m˜(i)F and n˜(i)F are, respectively, the slip direction vector and the vector normal to the slip plane measured in the current configuration, defined by
˜
m(i)F = Fem(i)F and n˜ (i)
F = F
−T
e n(i)F . (2.6)
Clearly, from (2.6), the vectorsm˜(i)F andn˜(i)F are not unit vectors.
2.1.2 Thermodynamic formulations
Decomposition of entropy density
The objective of formulating the model in a thermo-mechanical framework is to derive a consistent expression of the driving force for plastic slip. In thermo-mechanical processes, the entropy and temperature may be viewed as the thermal analogues of deformation and stress, respectively [22, 143]. Hence, in analogy to the decomposition of the total deformation gradient in (2.1), the total entropy density per unit mass,η, is decomposed as
η = ηe+ ηp, (2.7)
whereηerepresents the conservative (reversible) part of the entropy density and
ηp is the entropy density related to the plastic deformation process. Similar to
the entropy decomposition in the isotropic elasto-plasticity model of Simo and Miehe [116], the rate of change of the plastic entropy is assumed to be propor-tional to the rate of change of the plastic deformation, which here is measured by the rate of slip ˙γ(i)F ,
˙ηp= NF
X
i=1
˙γ(i)F φ(i)F , (2.8)
whereφ(i)F is interpreted as the entropy density related to plastic deformation per unit slip in systemi.
Balance principles and dissipation
Let P be the first Piola-Kirchhoff stress in the reference configuration, bf the
body force per unit reference volume and a the acceleration of a material point x. Assuming that all variables are continuously differentiable, the balance of linear momentum per unit volume in the reference configuration is given by
div P + bf = ρ0a , (2.9)
withρ0the ferrite mass density in the reference configuration.
Furthermore, letǫ be the internal energy density per unit mass, q the heat flux
per unit reference area andr the body heat source per unit reference volume. The
balance of total energy, combined with the balance of linear momentum per unit reference volume can be expressed as
ρ0˙ǫ + (div q− r) − P · ˙F = 0 , (2.10)
where the term P · ˙F is known as the internal power.
The rate of change of entropy per unit volume in the reference configuration,
Γ, is defined by
Γ := ρ0˙η + div Φ− s , (2.11)
where Φ and s are, respectively, the entropy flux per unit area and the entropy
source per unit volume in the reference configuration,
Φ= q
θ and s =
r
θ , (2.12)
withθ the (absolute) temperature. Defining the dissipation density per unit
refer-ence volume asD := Γθ and invoking equations (2.10)-(2.12), the total dissipa-tion can be written as
D = −ρ0˙ǫ + ρ0θ ˙η + P · ˙F − ∇θ · Φ . (2.13)
Taking the time derivative of the total deformation gradient in (2.1) and combining it with the expression for the plastic velocity gradient given in (2.3), provides the following expression for the internal power:
P · ˙F = P FpT · ˙Fe+ NF
X
i=1
where τF(i) is referred to as the resolved shear stress (or Schmid stress) for slip systemi, given by
τF(i) := FeTP FpT ·m(i)F ⊗ n(i)F . (2.15) From expressions (2.7) and (2.8), the contribution of the term ρ0θ ˙η to the total
dissipation can be obtained as
ρ0θ ˙η = ρ0θ ˙ηe+ NF
X
i=1
ζF(i)˙γF(i), (2.16)
where the quantityζF(i)in (2.16) can be interpreted as the thermal analogue of the resolved shear stress, as given by
ζF(i) := ρ0θφ(i)F . (2.17)
Subsequently, the rate of change of the internal energy density ˙ǫ that appears
in (2.13) needs to be determined. The internal energy density in the present model is decomposed into various mechanical and thermal contributions: The bulk strain energy density is characterized by the elastic deformation gradient Fe while the
thermal energy density is dependent of the conservative entropyηe. Furthermore,
a scalar variableβF is introduced to represent the local strains (or distortions) of
the BCC ferrite lattice associated with the presence of dislocations. Correspond-ingly, a lower length scale strain energy, called the lattice defect energy, can be expressed as a function of the scalar microstrainβF. The internal energy density
ǫ is assumed to be dependent of the state variables Fe,ηeandβF. In accordance
with the Coleman and Noll procedure [29], it is momentarily assumed that the internal energy densityǫ also depends on the fluxes ˙βF and Φ, i.e.,
ǫ = ¯ǫ(Fe, ηe, βF; ˙βF, Φ) . (2.18)
Using (2.14), (2.16) and (2.18), the expression for the total dissipation in (2.13) can be rewritten as D = P FpT − ρ0 ∂¯ǫ ∂Fe · ˙Fe+ ρ0 θ− ∂¯ǫ ∂ηe ˙ηe + NF X i=1
τF(i)+ ζF(i)˙γF(i)− ρ0
∂¯ǫ ∂βF ˙ βF − ρ0 ∂¯ǫ ∂ ˙βF ¨ βF − ∇θ · Φ − ρ0 ∂¯ǫ ∂Φ · ˙Φ . (2.19)
The second law of thermodynamics requires that the local entropy rate must be non-negative during any thermo-mechanical process, i.e.,Γ≥ 0. This restriction
leads to a non-negative energy dissipation, i.e. D ≥ 0, since the (absolute) tem-peratureθ is strictly positive. Furthermore, the terms in (2.19) that are multiplied
with the rates ˙Fe, ˙ηe, ¨βF and ˙Φ must vanish since otherwise a process could be
specified for which the dissipation is negative. This requirement results in
P = ρ0 ∂¯ǫ ∂Fe Fp−T and θ = ∂¯ǫ ∂ηe , (2.20)
and that the internal energy densityǫ does not depend on the fluxes ˙βF and ˙Φ.
In anticipation of a constitutive model for hardening and in order to simplify the presentation, the rate of change of the scalar microstrain, ˙βF, is taken to be
linearly dependent of the rate of change of the plastic slip, ˙γF(i), as follows:
˙ βF = NF X i=1 wF(i)˙γF(i), (2.21)
where the functions w(i)F depend on the slip resistance, as will be discussed in Section 2.1.5. From (2.20) and (2.21), the remaining non-zero terms of the total dissipation in (2.13) can be decomposed into the dissipation related to the plastic deformation, Dp, and the dissipation due to the heat conduction process,Dq, i.e.
D = Dp+Dq, (2.22)
whereDpandDqare, respectively, given by
Dp:= NF X i=1 τF(i)+ ζF(i)− ρ0 ∂¯ǫ ∂βF wF(i) ˙γF(i) and Dq:=−∇θ · Φ . (2.23)
Following the formalism proposed by Onsager for irreversible thermodynamics (see e.g., [2, 22]), for each physical phenomenon where the energy is dissipated, a pair of conjugate quantities can be identified as an affinity (or driving force) and the corresponding flux. In relation to the dissipation due to plastic deformation,
Dp, the driving force for plastic slip along a systemi is defined by
g(i)F := τF(i)+ ζF(i)− ρ0
∂¯ǫ ∂βF
whereas the rate of plastic slip ˙γF(i) is viewed as the corresponding flux. In re-lation to the dissipation due to heat conduction, Dq, the affinity is given by the
temperature gradient, −∇θ, with Φ as the corresponding flux. In view of the decomposition of the total dissipation (2.22), the dissipation inequality can be written as
D = Dp+Dq≥ 0 . (2.25)
In the present model, it is assumed that the dissipation inequality holds for the plastic deformation and heat conduction processes independently, which results in
Dp≥ 0 and Dq≥ 0 . (2.26)
2.1.3 Constitutive relations and Helmholtz energy density
It is common practice to work with the Helmholtz energy density ψ instead of
the internal energy density ǫ in order to use the temperature as an independent
variable instead of the entropy. The Helmholtz energy density, which depends on the state variables Fe,θ and βF, can be obtained from the internal energy density
using the following Legendre transformation1:
¯
ψ(Fe, θ, βF) = ¯ǫ(Fe, ¯ηe(Fe, θ, βF), βF)− θ¯ηe(Fe, θ, βF) . (2.27)
Relations between partial derivatives of the Helmholtz and the internal energy densities can be obtained by taking derivatives in (2.27) while holding the corre-sponding natural variables fixed, which results in
∂ ¯ψ ∂Fe = ∂¯ǫ ∂Fe , ∂ ¯ψ ∂θ =−ηe and ∂ ¯ψ ∂βF = ∂¯ǫ ∂βF , (2.28)
where the relation (2.20)2 was used to obtain (2.28)2.
In order to fulfil the principle of material frame indifference, the Helmholtz energy density (and also the internal energy density) cannot be dependent of the full elastic deformation gradient Fe. Alternatively, the elastic Green-Lagrange
strain Ee, which is based on the elastic stretch part only, is used instead. The
elastic Green-Lagrange strain is defined by
Ee:= 1
2 F
T
e Fe− I , (2.29)
1
Note that on the right hand side of (2.27), it uses the product θηeinstead of the more classical
with I the second-order identity tensor. Correspondingly, an alternative expres-sion for the Helmholtz energy density in terms of the elastic Green-Lagrange strain Eeis introduced such that
˜
ψ(Ee, θ, βF) = ¯ψ(Fe, θ, βF) . (2.30)
From the definition of the elastic Green-Lagrange strain in (2.29) and using the chain rule, the relation (2.28)1becomes
Fe ∂ ˜ψ ∂Ee = ∂ ¯ψ ∂Fe , (2.31)
where the symmetry of the elastic Green-Lagrange strain Ee was used.
Hence-forth, it is assumed that the Helmholtz energy density ˜ψ can be written as ˜
ψ(Ee, θ, βF) = ˜ψm(Ee) + ˜ψth(θ) + ˜ψd(βF) , (2.32)
where ˜ψm, ˜ψthand ˜ψd represent the contribution of the bulk strain energy,
ther-mal energy density and the lattice defect energy, respectively. Note that in the decomposition of the Helmholtz energy density (2.32), the terms ˜ψm, ˜ψthand ˜ψd
are fully decoupled.
Stress-elastic strain constitutive relation
Let S be the second Piola-Kirchhoff stress in the intermediate (or relaxed) con-figuration, which is related to the first Piola-Kirchhoff stress P measured in the reference configuration by
S= Fe−1P FpT . (2.33)
Note that the relation in (2.33) is derived by taking into account the fact that
Jp := det Fp = 1. From (2.20)1, (2.28), (2.31) and (2.33), the partial derivative
of the Helmholtz energy density with respect to the elastic Green-Lagrange strain is obtained as ∂ ˜ψ ∂Ee = 1 ρ0 S . (2.34)
The second Piola-Kirchhoff stress in the intermediate configuration, S, and its work-conjugated strain measure, the elastic Green-Lagrange Ee, are related
constitutively by
where CF is the fourth-order elasticity tensor of the BCC ferrite. In terms of a
commonly used6× 6 matrix representation (Voigt’s notation), the components of
the elasticity tensor CF can be written as
[CF]F = κF 1 κF2 κF2 κF2 κF1 κF2 κF2 κF2 κF1 κF 3 κF3 κF 3 F , (2.36)
whereκF1,κF2 andκF3 are the elastic moduli of the BCC ferrite. The subindexF
in (2.36) indicates that the stiffness components of CF are referred to the BCC
ferrite lattice basis. In the present model it is assumed that the elasticity tensor
CF does not depend on the elastic strain Ee nor on the temperature θ. Hence,
using the stress-strain constitutive relation (2.35) and through integrating the par-tial derivative (2.34) with respect to Ee, the expression of the bulk strain energy
density ˜ψm(Ee) can be written as
˜
ψm(Ee) =
1 2ρ0
CFEe· Ee. (2.37)
Reversible entropy-temperature constitutive relation
Similar to the stress and elastic strain relation, the reversible part of the entropy densityηeis related constitutively to the temperatureθ as follows [136, 143]:
ηe= hFln
θ θF
+ ηF , (2.38)
wherehF is the specific heat of the BCC ferrite, which, in this case, is assumed to
be a constant (temperature-invariant), andθF andηF are, respectively, a reference
temperature and a reference entropy density. In view of (2.38), the derivative of the Helmholtz energy density with respect to temperature in (2.28)2is given by
∂ ˜ψ ∂θ =−hFln θ θF − ηF . (2.39)
Through integrating (2.39) with respect to temperature, the thermal energy density ˜ ψth(θ) can be written as ˜ ψth(θ) =−hFθ ln θ θF + (hF − ηF) θ . (2.40)
Lattice defect energy density (cold work)
In addition to the bulk strain energy density, a lower scale elastic strain energy density is introduced that accounts for the (elastic) distortion of the lattice due to the presence of dislocations. This energy density term is called lattice defect
energy or cold work. The present model does not explicitly resolve the kinematics
and kinetics at the length scale of individual dislocations. Instead, an isotropic phenomenological model that is commonly used in the materials science literature is adopted. According to this model, the elastic strain energy associated with a single dislocation is proportional to µb2, where µ is an equivalent (isotropic)
shear modulus andb is the magnitude of the Burger’s vector (see e.g., Hull and
Bacon [55]). Further, the expression for the defect energy per unit volume is given by 12ωµb2ρd, whereρdmeasures the total dislocation line per unit volume andω
is a scaling factor for strain energy of an assembly of dislocations. For notational convenience, it is useful to introduce a strain-like internal variable, i.e.β := b√ρd
(see also [27]).
Adopting the above model, the lattice defect energy per unit mass, ψd, is
defined as a function that depends quadratically on the microstrainβF, i.e.,
˜
ψd(βF) :=
1 2ρ0
ωFµFβF2 , (2.41)
whereωF andµF are, respectively, the scaling factor that accounts for an
assem-bly of dislocations and the equivalent isotropic shear modulus of the BCC ferrite lattice. The equivalent shear modulusµF can be determined in terms of the elastic
moduliκFj , withj = 1, 2, 3, following the averaging procedure outlined in [141],
which gives µF = 1 10 2 κ F 1 − κF2 + 3κF3 . (2.42)
(2.37), (2.40) and (2.41), the Helmholtz energy density per unit mass is given by ˜ ψ(Ee, θ, βF) = 1 2ρ0 CFEe· Ee+ 1 2ρ0 ωFµFβF2 − hFθ ln θ θF + (hF − ηF) θ . (2.43)
2.1.4 Driving force, non-glide stress and kinetic law
Using the expression of the Helmholtz energy density given in (2.43) and in view of (2.28)3, the driving force for plastic slip in systemi in (2.24) can be
reformu-lated as
gF(i)= τF(i)+ ζF(i)− ωFµFβFw(i)F , (2.44)
withτF(i)andζF(i)given by (2.15) and (2.17), respectively. Using (2.33),τF(i) can be written in terms of the second Piola-Kirchhoff stress as
τF(i)= FeTFeS·
m(i)F ⊗ n(i)F
. (2.45)
As mentioned earlier in this chapter, the non-planar spreading of the cores of
1
2h111i screw dislocations causes the slip along an individual plane of the {110}
class to become dependent on resolved stresses acting on or normal to the non-glide plane, i.e. another {110} plane of the [111] zone. In accordance with the model proposed by Bassani et al. [9] (see also, Vitek et al. [145]), the non-glide stress τˆF(i) corresponding to a slip system i is defined as a resolved shear stress
parallel to the slip direction acting on the non-glide plane, i.e.,
ˆ τF(i)= FeTFeS· m(i)F ⊗ ˆn(i)F , (2.46)
wherenˆ(i)F is the unit vector perpendicular to the corresponding non-glide plane. The choice of the non-glide plane for each slip systemi follows from the results
of atomistic simulations [36, 146], which determines the asymmetry of slips. For example, the slip system [111](01¯1) corresponds to the non-glide plane (¯110)
whereas the opposite slip system[¯1¯1¯1](01¯1) relates to the non-glide plane (10¯1).
The expression (2.46) for the non-glide stress is formally similar to the expression (2.45) for the Schmid stress, wherenˆ(i)F plays an equivalent role as n(i)F . In accor-dance with the model of Bassani et al. [9], the contributions of non-glide stresses acting perpendicular to the slip direction are not accounted for.
The effect of the non-glide stress on the evolution of plastic slip can be mod-eled by incorporating this term into the “effective” resistance against plastic slip,
ˆ
s(i)F = s(i)F − ˆa(i)ˆτF(i), (2.47) which includes the slip resistance s(i)F and the effect of the non-glide stressτˆF(i), withˆa(i)a factor that determines the net contribution of the non-glide stress to the
“effective” slip resistance.
In a rate-dependent crystal plasticity formulation, the evolution of plastic slip in a slip systemi is described using a kinetic law, which relates the driving force gF(i) to the rate of slip ˙γF(i). The kinetic law must be defined such that it satisfies the requirement of non-negative energy dissipation. In the present model, the power law kinetic relation proposed by Cuiti˜no and Ortiz [30] is adopted, i.e., (see also [88]) ˙γF(i)= ˙γF 0 g(i)F ˆ s(i)F !(1/pF) − 1 ifg (i) F > ˆs (i) F , 0 otherwise , (2.48) where ˙γF
0 andpF are, respectively, the reference slip rate and the rate-sensitivity
exponent. Both parameters have positive values. The kinetic law (2.48) will re-duce to a rate-independent model as ˙γF
0 → ∞ and/or pF → 0 (see Figure 2.2).
Notice that the above kinetic relation gives a distinction between the elastic and plastic regimes explicitly. Furthermore, the power law equation (2.48) always leads to a non-negative plastic slip rate so that positive and negative senses of slip are accounted for separately. Plastic slip is initiated as soon as the driving force of a slip system exceeds a critical value, i.e.,
gF(i)≥ ˆs(i)F . (2.49)
2.1.5 Hardening and evolution of microstrain
In general, the magnitude of the slip resistance s(i)F evolves during plastic de-formations, which is defined through a hardening model. In the present work,
(b) (a)
Plastic driving force gF(i)
Rate of plastic sli p γF (i ) . Rate of plastic sli p γF (i ) .
Plastic driving force gF(i) gF(i) = s F (i) ˆ gF(i) = s F (i) ˆ increasing γ0 . decreasing pF F γ0 → ∞ .F pF → 0
Figure 2.2: Rate of change of plastic slip as a function of plastic driving force according to the kinetic relation (2.48) with variations of (a) reference slip rate and (b) rate-sensitivity exponent.
the evolution of the slip resistances(i)F is computed using the phenomenological model proposed by Peirce et al. [101],
˙s(i)F =
NF
X
j=1
HF(i,j)˙γF(j), (2.50)
whereHF(i,j)is a matrix containing the hardening moduli with the diagonal terms referring to self-hardening and the off-diagonal terms referring to cross-hardening, i.e., HF(i,j)= ( k(j)F fori = j , qFk(j)F fori6= j . (2.51) Here,qF defines the ratio between cross- and self-hardening moduli on each slip
system, called the latent hardening ratio, and k(j)F is the single-slip hardening modulus of a slip systemj. The evolution law for the single-slip hardening
mod-ulus is described by a power law equation proposed by Brown et al. [20], i.e.,
k(j)F = kF0 1−s (j) F sF ∞ !uF , (2.52) withkF
0 a reference hardening modulus,sF∞the saturation value of the slip
In addition, the initial value of the slip resistances(i)F is given by
s(i)F (t = 0) = sF0 . (2.53)
In the present model, the initial value for the slip resistance,sF0, is assumed to be the equal for all slip systems.
When introducing the microstrain variable βF in Section 2.1.2, it was
as-sumed beforehand that the rate of change of the microstrain parameter was con-nected to the rate of change of plastic slip through the functionsw(i)F (c.f., equation (2.21)). In line with the model proposed by Clayton [27], the state variable βF
is constitutively related to the average value of the slip resistance, which in a rate form can be written as
cFµFβ˙F = 1 NF NF X i=1 ˙s(i)F , (2.54)
withcF a scaling factor that accounts for average hardening. The assumption of
isotropy in (2.54) is adopted for reasons of simplicity. Substituting (2.50) into (2.54) leads to the following expression for the rate of change of the microstrain
˙ βF: ˙ βF = 1 cFµFNF NF X i=1 NF X j=1 HF(i,j)˙γF(j). (2.55)
The functions w(i)F can be related to the hardening moduli matrixHF(j,i)by com-paring the expressions (2.55) and (2.21), which results in
w(i)F = 1 cFµFNF NF X j=1 HF(j,i). (2.56)
Summary of single crystal elasto-plasticity model for ferrite
For convenience, the main ingredients of the elasto-plasticity model for single-crystalline ferrite are summarized as follows: The decompositions of the defor-mation gradient (2.1) and the entropy density (2.7) are, respectively,
The evolution of the plastic parts of the deformation gradient, Fp, and of the
entropy density,ηp, are, respectively, described by
Lp = ˙FpFp−1 = NF
X
i=1
˙γF(i)m(i)F ⊗ n(i)F and ˙ηp= NF
X
i=1
˙γF(i)φ(i)F .
The constitutive relations between conjugated variables, i.e., stress-elastic strain (2.35) and temperature-reversible entropy (2.38), are
S = CFEe and ηe= hFln
θ θF
+ ηF .
The relation between plasticity driving force and the rate of plastic slip (kinetic relation) is given by ˙γF(i) = ˙γ0F gF(i) ˆ s(i)F !(1/pF) − 1 ifg (i) F > ˆs (i) F , 0 otherwise ,
where the driving force for plastic slip gF(i) includes the contributions of the re-solved shear stress (Schmid stress), the plastic entropy density and the defect en-ergy, i.e,
g(i)F = FeTFeS·
m(i)F ⊗ n(i)F + ρ0θφF(i)− ωFµFβFw(i)F ,
and the “effective” slip resistance sˆ(i)F = s(i)F − ˆa(i)τˆ(i)
F , which accounts for the
contribution of the “classical” slip resistance and the effect of the non-glide stress
ˆ
τF(i)= FeTFeS·
m(i)F ⊗ ˆn(i)F .
Finally, the evolution laws for the slip resistance s(i)F (hardening model) and the microstrainβF are, respectively, given by
˙s(i)F = NF X j=1 HF(i,j)˙γF(j) and β˙F = NF X i=1 w(i)F ˙γF(i),
[100]F-loaded [110]F-loaded [111]F-loaded e1F e2F e3F e1F e2F e3F e1F e2F e3F f1 f2 f3Global axis (0˚,0˚,0˚) (45˚,0˚,0˚) (45˚,35.26˚,0˚)
Figure 2.3: Schematic representation of the crystallographic orientation of the ferrite single crystal samples with respect to the global basis{f1, f2, f3}.
2.2
Simulations of elasto-plastic deformation of
single-crystalline ferrite
In order to illustrate the basic features of the crystal plasticity model for BCC ferrite, the mechanical behavior of a single crystal ferritic sample is studied by means of numerical simulations. In the present work, three elementary loading modes are considered, namely (i) uniaxial tension and compression, (ii) simple shear and (iii) plane-stress equibiaxial stretch. Furthermore, the analyses are carried out considering three different crystallographic orientations, which, ex-pressed in terms of the “323”-Euler rotation (about the global basis with cartesian
unit vectors{f1, f2, f3}), are (0◦, 0◦, 0◦), (45◦, 0◦, 0◦) and (45◦, 35.26◦, 0◦),
re-spectively, The above orientations are chosen such that the global f1-axis
corre-sponds to, respectively, the[100]F,[110]F and[111]F directions, where the Miller
indices refer to the basis of the BCC lattice, as illustrated in Figure 2.3.
2.2.1 Material parameters and validation
The parameters used in the crystal plasticity model with the non-glide stress ef-fect for the BCC ferrite are discussed in this section. The elastic moduli for the BCC ferrite used in (2.36) are obtained from the data reported in Kurdju-mov and Khachaturyan [73], i.e., κF
1 = 233.5, κF2 = 135.5 and κF3 = 118.0
[GPa]. With these data, the equivalent (isotropic) shear modulus is obtained from (2.42) asµF = 55.0 GPa. The mass density of the ferrite (in the reference
car-0 200 400 600 800 1200 1000 Axi al Cauc hy st re ss T11 [M P a] 0 0.025 0.050 0.075 0.100 0.125 0.150
Axial nominal strain ε11 Polycrystal model (Taylor average) Experimental data
(75% ferrite + 25% bainite)
Figure 2.4: Axial stress-strain response of a Taylor-type polycrystalline sample fitted to experimental data of Jacques et al. [63] for ferrite-based material.
bon steel, ρ0 = 7800 kg·m−3. For simplicity, it is assumed that the values of
the weight parameters for the non-glide stress contribution used in (2.47) are the same for all slip systems, i.e,ˆa(i)= ˆa. The value for ˆa is calibrated from the data
of uniaxial tensile tests on single crystal BCC ferrite presented in [40], particu-larly, the data on the asymmetry of the resolved shear stressτ on theˆ {110} and {211} (twinning and antitwinning) planes. Following the procedure highlighted
in Bassani et al. [9], the magnitude ofˆa is obtained by fitting the following curve: ˆ
τ = ˆτcr/[cos φ + ˆa cos(φ + 60◦)] to the experimental data [40]. In this case, ˆτcr
is the critical resolved shear stress on the maximum resolved shear stress plane (MRSSP), the angle φ defines the orientation of the MRSSP with respect to the
slip plane anda is a parameter that characterizes the asymmetry of ˆˆ τ in the
twin-ning and antitwintwin-ning senses. This calibration procedure results inˆa = 0.12.
Furthermore, the parameters for the power-law kinetic model (2.48), i.e., ˙γ0F
andpF, are chosen such that the overall response under quasi-static loading
con-ditions is close to a rate-independent response. The purpose of introducing a small rate-dependency is to avoid numerical singularity problems often encountered in rate-independent crystal plasticity models [88]. For this reason, the parameters for the kinetic model are taken as ˙γF
0 = 0.001 s
−1
and pF = 0.02, which fall
within the typical range of values used in rate-dependent crystal plasticity models (see, e.g., [30, 88]).
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